Factoring Deferred Values by Survival to Deferment

Why Factor a Deferred Benefit?

A deferred death benefit pays nothing if death occurs during an initial waiting period. Its present value therefore depends on two distinct stages: reaching the end of that period alive, and then receiving the insurance benefit if death occurs later. Separating these stages makes both the calculation and its assumptions easier to check.

The factorization applies to continuous death benefits: the benefit is paid at the moment of death. It expresses the value at issue as the probability of surviving to the deferment date, adjusted for discounting to that date, multiplied by the insurance value at that date conditional on being alive. The conditional value must use mortality and interest assumptions applying after deferment.

Definition: Suppose a unit death benefit is payable only if death occurs after $n$ years. The deferment factor is the probability of surviving $n$ years multiplied by the discount factor from issue to year $n$. The conditional value is the expected present value, measured at year $n$, of the benefit given survival to year $n$. For a deferred whole-life benefit, this conditional value is $\bar{A}_{x+n}$; for a deferred term benefit, it is the value of the specified term insurance for a life aged $x+n$.

The Factorization

Assume a life aged $x$ is subject to a continuous mortality model with force of mortality $\mu_{x+t}$ at duration $t$, and the force of interest is a constant $\delta$. Then ${}_t p_x$ is the probability of survival for $t$ years, and $e^{-\delta t}$ discounts a payment at time $t$ to issue. A unit death benefit paid after $n$ years has present value

\int_n^\infty e^{-\delta t}{}_t p_x\mu_{x+t}\,dt = e^{-\delta n}{}_n p_x \int_0^\infty e^{-\delta s}{}_s p_{x+n}\mu_{x+n+s}\,ds.

The integral on the left is the direct value: for death at time $t$, discount the benefit to issue and multiply by the density of death at $t$, which is ${}_t p_x\mu_{x+t}$. On the right, the first two factors account for surviving and discounting to time $n$; the remaining integral is the value from time $n$ onward, conditional on being alive at that time.

Theorem: For a unit continuous deferred whole-life benefit with deferment $n$, the value at issue is $e^{-\delta n}{}_n p_x\bar{A}_{x+n}$, where $\bar{A}_{x+n}$ is the continuous whole-life insurance value for a life aged $x+n$ under the post-deferment assumptions. For a unit continuous term benefit payable only for deaths from time $n$ through time $n+m$, the value is $e^{-\delta n}{}_n p_x$ multiplied by the conditional $m$-year continuous term insurance value at age $x+n$.

The result follows by setting $t=n+s$ in the direct integral. Survival factors obey ${}_{n+s}p_x={}_n p_x\,{}_s p_{x+n}$. Also, discounting to time $n+s$ separates as $e^{-\delta(n+s)}=e^{-\delta n}e^{-\delta s}$. Substituting these two identities into the integral leaves the common factor $e^{-\delta n}{}_n p_x$ outside, with the remaining integral representing the conditional value after deferment. This is a change of time origin and a conditional-survival decomposition, not an assumption that mortality after deferment matches mortality before it.

If mortality and interest forces are constant after deferment, write their values as $\mu$ and $\delta$. For a unit whole-life benefit, the conditional value is

\int_0^\infty e^{-(\mu+\delta)s}\mu\,ds = \frac{\mu}{\mu+\delta}.

For a unit term benefit lasting $m$ years after deferment, the corresponding conditional value is

\int_0^m e^{-(\mu+\delta)s}\mu\,ds = \frac{\mu}{\mu+\delta}\left(1-e^{-(\mu+\delta)m}\right).

These closed forms require constant mortality and interest forces over the conditional period. If either force varies, use the appropriate integral or value under the specified post-deferment model instead. The deferment factor itself must also reflect the mortality and interest assumptions during the initial $n$ years.

Worked Applications

Worked Example: Constant-Force Deferred Whole Life

A hypothetical policy pays a unit benefit at death, but only if death occurs more than two years after issue. Assume a constant mortality force of $0.03$ per year and a constant interest force of $0.05$ per year, both before and after deferment. Find the value at issue, to four decimal places.

Method. Factor the value into survival and discounting for two years, followed by the conditional whole-life value. The common trap is to discount to year two but forget the probability of surviving to year two.

The survival probability to year two is

{}_2p_x=e^{-0.03(2)}=e^{-0.06}=0.9417645.

The discount factor to year two is

e^{-0.05(2)}=e^{-0.10}=0.9048374.

Thus the deferment factor is $0.9417645\times0.9048374=e^{-0.16}=0.8521438$. Conditional on survival to year two, the whole-life value is

\bar{A}_{x+2}=\frac{0.03}{0.03+0.05}=0.3750.

Multiplying gives $0.8521438\times0.3750=0.3195539$, or 0.3196 to four decimal places.

Check. Direct integration from issue gives $\int_2^\infty 0.03e^{-0.08t}\,dt=(0.03/0.08)e^{-0.16}=0.3195539$, agreeing with the factorized result.

Worked Example: Deferred Term Insurance With a Finite Coverage Window

A hypothetical unit continuous term benefit is payable for deaths occurring from year four through year six after issue. Assume constant mortality force $0.04$ per year and constant interest force $0.06$ per year throughout. Find the value at issue, to four decimal places.

Method. The deferment is four years, and the term after deferment lasts two years. First calculate the survival-and-discount factor to year four; then calculate the two-year term value conditional on being alive at year four. Do not treat the coverage as a six-year term beginning at issue.

The deferment factor is

e^{-(0.04+0.06)4}=e^{-0.40}=0.6703200.

For the conditional two-year term, the combined force $\mu+\delta$ is $0.04+0.06=0.10$. Its value at year four is

\frac{0.04}{0.10}\left(1-e^{-0.10(2)}\right) =0.4(1-e^{-0.20}) =0.4(1-0.8187308) =0.0725077.

The value at issue is therefore

$$ 0.6703200(0.0725077)=0.0486034. $$

To four decimal places, the value is 0.0486.

Check. Directly, integrate the death-payment density only over years four to six: $\int_4^6 0.04e^{-0.10t}\,dt=0.4(e^{-0.40}-e^{-0.60})=0.0486034$. This agrees with the factorization and confirms that the two-year coverage window begins at year four.

Worked Example: Piecewise Mortality Before and After Deferment

A hypothetical policy pays $ \$20{,}000$ at death after a three-year deferment. The mortality force is $0.01$ during the first year and $0.05$ during the next two years. After year three, the mortality force is $0.03$ for life. The interest force is $0.02$ per year throughout. Find the value at issue to the nearest cent.

Method. Calculate survival to year three using the two pre-deferment mortality forces, and discount to year three using the interest force for all three years. Then multiply by the conditional whole-life value using the post-deferment mortality force. The common mistake is to omit the three years of discounting, or to use the post-deferment mortality force to calculate earlier survival.

The accumulated mortality force before deferment is $0.01(1)+0.05(2)=0.11$. Hence

{}_3p_x=e^{-0.11}=0.8958341.

Discounting for three years gives $e^{-0.02(3)}=e^{-0.06}=0.9417645$. The deferment factor is therefore

e^{-0.11}e^{-0.06}=e^{-0.17}=0.8436648.

After year three, the conditional whole-life value is

\bar{A}_{x+3}=\frac{0.03}{0.03+0.02}=0.6000.

For a unit benefit, the value at issue is $0.8436648(0.6000)=0.5061989$. Scaling by the $ \$20{,}000$ benefit gives

\$20{,}000(0.5061989)=\$10{,}123.98.

Check. The result is less than the benefit amount, as a present value should be. Its factorization uses the first-year and next-two-year mortality assumptions only for survival to deferment, and uses the later mortality force only for the conditional whole-life value.

Conditions, Interpretation, and Exam Traps

The factorization is useful whenever a benefit begins only after survival to a specified time. It reduces a deferred calculation to two interpretable parts and can simplify models in which the mortality assumptions change at deferment. It also provides a check: the deferment factor is a probability multiplied by a discount factor, and therefore is no greater than one for nonnegative interest and a valid survival model.

The benefit must not be payable during the deferment period for the simple product to represent the entire contract value. If a contract also pays a benefit on death during deferment, that early-death value must be calculated separately and added. For a term benefit, identify both endpoints correctly: a deferment of $n$ years followed by $m$ years of coverage corresponds to deaths from time $n$ through time $n+m$, not a term of length $n+m$ starting immediately.

Omitting pre-deferment survival: using only $e^{-\delta n}$ incorrectly assigns value to policies whose insured dies before coverage begins. For Example 1, this error gives $e^{-0.10}(0.3750)=0.3393$, rather than the correct $0.3196$.
Omitting pre-deferment discounting: survival alone is not the value at issue of reaching a future date; the value must also be discounted back to issue.
Using the wrong mortality period: survival to deferment uses the mortality assumptions before deferment, while the conditional insurance value uses the assumptions afterward.
Confusing duration and age: the conditional value is for a life aged $x+n$, with time measured from the deferment date. Its discounting is relative to that date; the factor to discount from the deferment date back to issue is already included outside it.

Key takeaway: For a deferred continuous death benefit, multiply the value conditional on survival to deferment by both ${}_n p_x$ and the discount factor to time $n$. Keep the pre-deferment and post-deferment assumptions in their respective parts of the calculation.

Check Your Understanding

For each problem, identify the deferment factor and conditional value before multiplying.

A unit whole-life benefit is deferred for one year. Mortality and interest forces are constant at $0.02$ and $0.03$, respectively, for all time. Find the value at issue to four decimal places. (Answer: $0.3805$)
A unit continuous term benefit starts after two years and lasts for three years. Constant mortality and interest forces are $0.05$ and $0.05$. Find the value at issue to four decimal places. (Answer: $0.1061$)
A unit whole-life benefit is deferred for two years. Mortality force is $0.02$ in year one and $0.04$ in year two, then $0.05$ for life. The interest force is $0.03$ throughout. Find the value at issue to four decimal places. (Answer: $0.5543$)
A $ \$15{,}000$ whole-life benefit is deferred for two years. Mortality and interest forces are constant at $0.04$ and $0.06$, respectively. Find the value at issue to the nearest cent. (Answer: $\$4{,}912.38$)

What You'll Learn